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Streaming Statistics in Constant Memory

"Compute the rolling mean and variance of a stream, in constant time per update" is a question research and developer interviews ask almost word for word. The obvious answer recomputes the statistics over the whole window for every new value, and on a tick stream that is far too slow: a window of 20,000 prices recomputed on every tick is 20,000 operations per tick. The answer interviewers want keeps a few running numbers and updates them as values enter and leave.

A growing window: running sums

When every value stays in the window, keep the count nn, the sum S=∑xiS = \sum x_i and the sum of squares Q=∑xi2Q = \sum x_i^2. Each new value adds to all three in O(1)O(1), and the statistics follow from them:

xˉ=Sn,s2=nQ−S2n(n−1)\bar{x} = \frac{S}{n}, \qquad s^2 = \frac{nQ - S^2}{n(n - 1)}

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